ASYMPTOTICAL EQUIVALENCE OF SOLUTIONS IN THE THEORY OF DYMAMICAL SYSTEMS

Abstract

In the paper, the following equivalence relations in spaces of solutions of initial value problems for dynamical systems are proposed. The asymptotical equivalence relation: distance between two solutions tends to zero while time increases, the corresponding quotient space was called “asymptotical quotient space“. The asymptotical exponential equivalence relation: distance between two solutions decreases exponentially while time increases, the corresponding quotient space was called “asymptotical exponential quotient space“. The Hausdorff asymptotical equivalence relation: distance between two solutions with invertible transformation of argument tends to zero while time increases; the corresponding quotient space is called “Hausdorff asymptotical quotient space“. It is demonstrated that the notion of the Hausdorff asymptotical quotient spaces generate new mathematical objects.